Replacing a subgroup by a normal subgroup

It often happens that we have a subgroup of a group satisfying certain properties, but we want a normal subgroup of satisfying the same, or similar, properties. One way of going about this is to try showing that itself is normal. However, this may be hard to establish, and may in fact be false. There are some other techniques that we can use to replace by a normal subgroup of .

For more on techniques to directly prove that a given subgroup is normal, refer to proving normality.

The normal core

General approach

The normal core of a subgroup in a group , denoted , is defined as the intersection of all conjugate subgroups of in . It is also the largest normal subgroup of contained in , and also equals the kernel of the homomorphism from to given by the left multiplication action on the left coset space.

Here are some important facts:

The normal core of is contained inside . Thus, if we are interested in preserving some property that is subgroup-closed, we can pass to the normal core. For instance, if is abelian, the normal core of is abelian. If is a solvable group, the normal core of is also solvable.

The normal core of is not much smaller than . Specifically, if has index , the index of the normal core of divides . This result is called Poincare's theorem, and it implies that if has finite index, so does the normal core of . Even in finite groups, this can be used to show that the normal core is a nontrivial normal subgroup, by showing that the index of the normal core is strictly smaller than the order of the group. The latter technique is called the small-index subgroup technique and is used in Sylow theory to show that groups of certain orders cannot be simple.

Thus, we can replace a subgroup by its normal core, while preserving subgroup-closed group properties, and not getting too much smaller in the sense that the index of the normal core is bounded in terms of the index of the subgroup.

The normal closure

The normal closure is a larger subgroup than the original subgroup, but there is no size bound on it in terms of the size of the original subgroup. Thus, it is not possible to guarantee, for instance, that the normal closure of a finite subgroup is finite, or even finitely generated. Also, since the normal subgroup is bigger, it need not satisfy any of the subgroup-closed group properties satisfied by the original subgroup. On the other hand, if a group property is closed under taking joins of groups, then the normal closure of a group satisfying it also satisfies it. For instance, the normal closure of a perfect subgroup is a perfect subgroup.

Abelian-to-normal replacement theorems

Replacement theorems are results that allow us to replace one subgroup in a group by another that is similar in some ways and better in others. There are a number of replacement theorems related to groups of prime power order, that guarantee the existence of normal subgroups similar to subgroups we have found. These are discussed below.

Below are given conditions under which, in a group of prime power order, we can guarantee that the existence of a subgroup of a certain type of order guarantees the existence of a normal subgroup of the same type and same order.

Threshold values

This lists threshold values of : the largest value of for which the collection of -groups of order satisfying the stated condition satisfies a weak normal replacement condition. The nature of all these is such that the weak normal replacement condition is satisfied for all smaller but for no larger . The between and below means that the minimum known value is and the maximum known value is .

Collection of groups

Abelian groups of order

between 4 and 5

between 5 and 13

between 5 and 6

between 5 and 7

between 6 and 9

between and

Abelian groups of order , exponent dividing

between 2 and 5

between 5 and 13

between 5 and 6

between 5 and 7

between 6 and 9

between and

Elementary abelian group of order

1

between 5 and 13 (?)

between 5 and 6 (?)

between 5 and 7

between 6 and 9

between and

Groups of exponent , order

1

at least 2

at least 4

at least 6

Other replacement theorems

There are some other replacement theorems in both the finite and infinite case. For instance: